Choppy movement with floats

When I move text slower than a pixel at a time, it doesn't move in subpixels, instead it jumps from pixel to pixel. How can I fix that?
I searched for days, still nothing that could help me with that.
Let me know if I need to show code of some part, I can upload everything, but it'll be a lot to fit in here.

Use a multisampled buffer. This will give you sub-pixel coverage, and smooth out the rasterization of your text. The higher the number of samples, the smoother it will become.

Alternatively, you can make the quad 1 pixel larger in each direction, compute the floating point position in screen coordinates where it should appear, and then compare that to the actual GL coords using gl_FragCoord.xy. Use the difference to adjust your texture coordinates.

I'm using GL_NEAREST, because GL_LINEAR is not what I'm looking for, it makes texture blurry. I want something like D3DTEXF_POINT (DX), it is smooth and have a nearest filtering. Exactly what I need for pixel art and stuff.
Thanks.

What I mean is: You are seeing an unsightly "pop" when your text image animates (moves). Because 1 pixel from the source image is being drawn at a particular destination pixel one second; then it is drawn in a neighboring destination pixel the next second. That is the "pop". You are moving the texture 1/100th of a pixel on every draw, but the results don't show until you have crossed a mathematical threshold.

That is the nature of nearest-neighbor sampling.
That is the point of it.
There is no sub-pixel distinction.
A source pixel can either live in one destination pixel, or it's neighbor. It can't live in both.

Living in both is what linear does.
(20% in the first pixel, 80% in the second pixel)